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  • View in gallery

    The Chesapeake Bay bathymetry and the tower and buoy locations for the field experiment: ★ tower location, ▴ CBOS buoy, and + Thomas Point Light tower

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    Layout of the temporary fixed tower for the experiment: (a) cross section and (b) plan view

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    The time series of (a) measured wind speeds and directions, (b) significant wave heights and mean wave directions, (c) peak wave periods, and (d) air and sea temperatures for the tower experiment from 19 to 28 Jul 1998. The lengths of the vectors are used to show the magnitude of the wind speed or significant wave height. The pointing directions of the vectors show wind directions or mean wave directions. The wind data in (a) are from CBOS raw wind data with the available air–sea flux data shown by dots under the raw wind vectors

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    Comparison of observed mean neutral drag coefficients for nine 1 m s−1 different wind speed bins with those calculated from Eq. (17) (dotted line) and Eq. (10) (solid squares). The data points are shown by open circles. Error bars represent the standard error for each wind speed bin

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    Linear regression between the measured neutral drag coefficients and the mean wind speeds, Un10, as compared with results from other field studies. Data points are shown by open circles. Lines from the top to the bottom are: Charnock (short dash), RASEX (dash-dotted), MARSEN (dotted), HEXMAX (long-dash), SWADE (dash-dotted-dotted), and regression from data by Eq. (18) (solid)

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    Nonlinear regression between measured neutral drag coefficient and wave age in Eq. (9). Data points are shown by open circles. Results from the present study (solid line) is in between those from MARSEN (dotted line) and RASEX (dash-dotted line)

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    Comparison of the measured neutral drag coefficients from the flux measurement (y axis) with those calculated by Eq. (11) (x axis) (Taylor and Yelland 2001). Solid circles: Cp/u∗ < 12, solid triangles: 12 < Cp/u∗ < 20, and open circles: Cp/u∗ > 20

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    An 11-h time series of (a) neutral wind speed, (b) wind direction, and (c) the drag coefficients. The Cd in (c) was calculated from flux measurements (open circles), from linear regression in Eq. (18) (solid lines), from the regression with wave age in Eq. (9) (solid triangles), and from Eq. (17) (dotted line). These Cd are plotted against the mean wind speeds Un10 in (d)

  • View in gallery

    (a) The wave model–predicted significant wave height Hs (y axis) vs the measured Hs > 10 cm (x axis) and (b) the model-predicted peak period Tp (y axis) vs measured Tp (x axis)

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    Comparison of the model-predicted drag coefficients for nine 1 m s−1 wind speed bins with the observed mean drag coefficients. Model-predicted Cd are shown by open triangles. The dash-dotted line with solid triangles shows the model-predicted bin-averaged Cd. The error bars represent standard errors for each bin. The bin-averaged data are shown by solid line with solid circles

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    The regression line calculated from the model-predicted drag coefficients (dash-dotted line) with the model-predicted wave age is compared with that from the measured data (solid line). Model-predicted drag coefficients are shown by open triangles

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Drag Coefficients with Fetch-Limited Wind Waves

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  • 1 Horn Point Laboratory, University of Maryland Center for Environmental Science, Cambridge, Maryland
  • | 2 Air Resource Laboratory, National Oceanic and Atmospheric Administration, Silver Spring, Maryland
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Abstract

Air–sea fluxes of momentum and heat were measured simultaneously with surface wind waves and near-surface currents in mid Chesapeake Bay during summer 1998 under low wind conditions. The data were collected using a Gill sonic anemometer and a Sontek Acoustic Doppler Velocimeter with a pressure sensor mounted on a temporary fixed tower in 8.8 m of water. The analyzed data show that the neutral drag coefficients depend upon both wind speed and wave age. They are better correlated to wave age than to wind speed. Data scatter is significantly larger in low winds than in high winds. Under light winds, the neutral drag coefficients increase with decreasing wind speed and have values much higher than those for relatively higher wind speeds. At higher wind speeds, neutral drag coefficients increase with increasing wind speed. Regardless of wind speed, neutral drag coefficients always decrease with increasing wave age. Neutral drag coefficients are lower than the results of similar field studies when fit to wind speed alone, but they statistically agree with other studies if they are fit to wave age. The momentum transfer mechanism is investigated using a parametric wave model with a sea-state-dependent form drag and a reference system moving with the waves. The relationship between modeled drag coefficient and modeled wave age agrees well with the relationship derived from the data.

+ Current affiliation: Oceanography Division, Naval Research Laboratory, Stennis Space Center, Mississippi

# Current affiliation: Integrated Forest Management, LLC, Sun Prairie, Wisconsin

Corresponding author address: Dr. Lawrence P. Sanford, Horn Point Laboratory, University of Maryland Center for Environmental Science, P.O. Box 775, Cambridge, MD 21613. Email: lsanford@hpl.umces.edu

Abstract

Air–sea fluxes of momentum and heat were measured simultaneously with surface wind waves and near-surface currents in mid Chesapeake Bay during summer 1998 under low wind conditions. The data were collected using a Gill sonic anemometer and a Sontek Acoustic Doppler Velocimeter with a pressure sensor mounted on a temporary fixed tower in 8.8 m of water. The analyzed data show that the neutral drag coefficients depend upon both wind speed and wave age. They are better correlated to wave age than to wind speed. Data scatter is significantly larger in low winds than in high winds. Under light winds, the neutral drag coefficients increase with decreasing wind speed and have values much higher than those for relatively higher wind speeds. At higher wind speeds, neutral drag coefficients increase with increasing wind speed. Regardless of wind speed, neutral drag coefficients always decrease with increasing wave age. Neutral drag coefficients are lower than the results of similar field studies when fit to wind speed alone, but they statistically agree with other studies if they are fit to wave age. The momentum transfer mechanism is investigated using a parametric wave model with a sea-state-dependent form drag and a reference system moving with the waves. The relationship between modeled drag coefficient and modeled wave age agrees well with the relationship derived from the data.

+ Current affiliation: Oceanography Division, Naval Research Laboratory, Stennis Space Center, Mississippi

# Current affiliation: Integrated Forest Management, LLC, Sun Prairie, Wisconsin

Corresponding author address: Dr. Lawrence P. Sanford, Horn Point Laboratory, University of Maryland Center for Environmental Science, P.O. Box 775, Cambridge, MD 21613. Email: lsanford@hpl.umces.edu

1. Introduction

Through transfer of momentum and sensible and latent heat, the turbulent atmospheric and oceanic boundary layers couple with each other across the wavy interface. This complex set of interactions is often expressed in terms of a drag coefficient (Cd) or an aerodynamic surface roughness (z0). Because of its central role in understanding and modeling air–sea interaction processes, Cd or z0 has been studied extensively from observations and numerical models in the past few decades. One important issue has been and continues to be the role of surface gravity waves in determining Cd or z0.

Several field experiments have reported evidence of wave age dependence in Cd or z0, for example, the Marine Remote Sensing (MARSEN) program (Geernaert et al. 1987), the Humidity Exchange over the Sea (HEXOS) program in the North Sea, the HEXOS Main Experiment (HEXMAX) off the Dutch coast (Smith et al. 1992), and the Risø Air Sea Experiment (RASEX) in Denmark (Mahrt et al. 1996; Vickers and Mahrt 1997). Other literatures (e.g., Kitaigorodskii 1968, 1970; Donelan et al. 1985; Donelan 1990; Merzi and Graf 1985; Toba et al. 1990) also suggested that wave age is an important parameter for determining Cd or z0. Most of these studies are based on data collected in fetch-limited conditions either in coastal waters or lakes. For open ocean data, however, a clear wave age dependence has not been reported (Dobson et al. 1994; Rieder 1997; Taylor and Yelland 2001). Therefore, a few studies (Smith and Banke 1975; Garratt 1977; Smith 1980; Large and Pond 1981) concluded that Cd or z0 depends on wind speed only.

Under steady state and horizontally homogeneous conditions, the wind profile in the atmospheric boundary layer can be described by the Monin–Obukhov (1954) similarity theory. The vertical wind profile is
i1520-0485-32-11-3058-e1
where Uz is the wind speed measured at anemometer height z, u∗ is the surface wind friction velocity and is given by (τ/ρa)1/2, τ is wind stress, κ is the von Kármán constant (=0.41 in this study), and L is the Monin–Obukhov stability length. It can be estimated by
i1520-0485-32-11-3058-e2
where g is the gravity, θ is the mean potential temperature in the boundary layer; θa is the air potential temperature and θs is the sea surface potential temperature. Here ψm (z/L) has been determined empirically (e.g., Liu et al. 1979; Large and Pond 1982; Panofsky and Dutton 1984; Erickson 1993; Ataktürk and Katsaros 1999). In the stable case (L > 0),
i1520-0485-32-11-3058-e3
and in the unstable case (L < 0),
i1520-0485-32-11-3058-e4
where x = (1 − 16z/L)1/4. The drag coefficient is defined as Cd = (u∗/Un10)2 and a unique relationship between Cd and z0 can be derived as Cd = [κ/ln(z/z0)]2 under neutrally stratified conditions.
Equation (1) is developed over land and is valid only under stationary and horizontally homogeneous conditions. Also, it cannot be applied very close to the surface where molecular transport is dominant and turbulence is suppressed. When applied in a marine environment, the over water atmospheric boundary layer must also meet the stationary and horizontally homogeneous conditions so that the surface layer can be treated as a constant stress layer. Under light wind conditions, the flow is aerodynamically smooth and viscous stresses dominate the momentum flux from air to the water surface. The surface roughness is parameterized with the friction velocity and the kinematic viscosity of air (e.g., Smith 1980; Large and Pond 1981; Donelan 1990; Banner et al. 1999):
i1520-0485-32-11-3058-e5
where νa = 1.4 × 10−5 m2 s−1 is the kinematic viscosity of air. The speed limit for a smooth flow condition is given by u∗ < 2(νag)1/3 (Donelan 1990) and can be estimated as u∗ = 0.103 m s−1, or Un10 ≈ 3.0 m s−1. The thickness of the viscous sublayer decreases with the increasing friction velocity. As wind speeds increase, the friction velocity increases and a windsea develops. A growing fraction of the momentum flux from air to the sea surface occurs as wave form drag. The surface makes a transition from a relatively smooth flow to a rough state with the waves penetrating through the viscous sublayer and interacting with the air flow. The surface roughness length increases and the airflow becomes aerodynamically rougher. For fully developed waves, Charnock (1955) proposed that z0 could be given as
i1520-0485-32-11-3058-e6
where α = 0.012 (Charnock 1958) is the Charnock constant. The values of α derived from data in the open ocean with well-developed waves do not agree with those obtained with fetch limited waves. For example, Smith (1980, 1988) and Large and Pond (1981) have shown that α = 0.011 best describes the dependence of Cd on wind speed observed over deep open ocean conditions. But applications to the datasets obtained in coastal waters or lakes with presumably less mature waves resulted in higher values of α, for example, α = 0.0145 by Garratt (1977) and α = 0.018 by Wu (1980).
Most of the field experiments over the past have established a statistically significant dependence of Cd on Un10. The general form of this linear regression can be expressed as
CdabUn10−3
where a and b are coefficients determined by the data. In MARSEN, a = 8.47 × 10−2 and b = 0.577 (Geernaert et al. 1987); in HEXMAX, a = 9.1 × 10−2 and b = 0.50 (Smith et al. 1992); in RASEX, a = 6.7 × 10−2 and b = 0.75 for Un10 > 4 m s−1 (Vickers and Mahrt 1997); and in Surface Wave Dynamics Experiment (SWADE), a = 7.0 × 10−2 and b = 0.6 for 6 m s−1 < Un10 < 14 m s−1 (Drennan et al. 1999). Banner et al. (1999) also selected quite a few observational results and showed a surprising large scatter among them. The review by Wu (1980) and Geernaert (1990) listed many different values of the coefficients for different observations.
It is clear that the variability of Cd not explained by Un10 is substantial. The surface waves must play a role in air–sea momentum fluxes. The wave form drag changes with wave age and in a fully developed wave field, the dominant waves traveling at phase speeds close to the wind velocity receive little momentum from the air (Dobson and Elliott 1978; Snyder et al. 1981; Hsiao and Shemdin 1983; Hasselmann et al. 1986). Steward (1974) recommended that a sea-state-dependent Charnock constant be used:
i1520-0485-32-11-3058-e8
where Cp is the phase velocity of the waves at the spectral peak. The form of the function in the right hand side of Eq. (8) can be different. A widely used function is
i1520-0485-32-11-3058-e9
where A and B are coefficients determined by the data. In MARSEN, A = 1.48 × 10−2 and B = −0.738 when long period swells are not included; in HEXMAX, A = 0.43 and B = −0.961; in RASEX, A = 7.1 × 10−3 and B = −2/3. However, Eqs. (8) and (9) may have a self-correlation problem if other variables, like Cp or u∗, have a limited range (Perrie and Toulany 1990; Smith et al. 1992).
Avoiding these problems, Donelan (1990) proposed a relationship between z0, Hs, and wave age Cp/Un10 (rather than Cp/u∗) for fully rough flow,
i1520-0485-32-11-3058-e10
based on data collected nearshore in Lake Ontario, Canada. Equation (10) is derived from data with fetch-limited waves and absence of swells or mixed seas. Similarly, Hsu (1974) suggested that Charnock constant is a function of wave steepness. Most recently, Taylor and Yelland (2001) combined the HEXMAX, RASEX, and Lake Ontario datasets (Anctil and Donelan 1996) and proposed that z0 be parameterized by the height and steepness of the waves,
i1520-0485-32-11-3058-e11
where Lp is the wave length at the spectral peak.

Despite a large number of field studies, controversy remains over the nature of the relationship among wind, sea state, and drag. A comprehensive reference and background knowledge on this subject have been given in Donelan (1990), Geernaert (1990), and more recently by Banner et al. (1999). In this paper, we present new data from an atmospheric boundary layer and surface wave study under almost purely fetch-limited conditions, with fetches ranging from a few to tens of kilometers (section 2). We present relationships between Cd, Un10 and Cp/u∗ derived from these data and compare the results with other observations (section 3). We also explore the mechanism underlying the results using a parametric wave model with a wave-age-dependent form drag and a wave-following reference frame (section 4). The model is a straightforward extension of the wave model developed by Donelan (1977). Finally, we summarize and conclude our study in section 5.

2. Experimental design and data analysis

a. Experiment layout

The field data used in this study were collected in the middle reaches of Chesapeake Bay (CB). The bay is a semienclosed basin and the largest estuary in the United States and stretches 320 km from the Susquehanna River to the Atlantic Ocean. It varies from 7 to 50 km in width and its average water depth is about 8.5 m. Surface waves in the mid CB are dominated by fetch-limited windseas (Sanford 1994; Lin et al. 1998; Lin 2000). Fetch varies strongly as a function of wind direction and location. Swell generated in the Atlantic Ocean can only affect the wave climate in the southern bay (Boon 1998). The swell energy is dissipated to undetectable levels by the time it reaches the mid CB. Thus, field experiments conducted in mid CB cover a wide range of wave ages under almost entirely fetch-limited conditions.

The field experiment was carried out from 19 to 28 July 1998. A temporary fixed tower was deployed at 38°29′41″N, 76°22′11″W on a shallow shelf adjacent to the deep center channel, approximately 3 km from the eastern shore and 10 km from the western shore (Fig. 1). The water depth at the tower site was 8.8 m with a mean tidal range of approximately 0.6 m. The tower height was 11 m. On top of the tower, a rectangular platform (1.8 × 1.2 m2) was mounted for deploying a navigation safety light, a Gill sonic anemometer and an infrared hygrometer (IRGA). The sonic anemometer and hygrometer were centered at 0.8 m above the platform and 3.8 m above mean water level. Surface wind waves and near-surface water velocities were measured simultaneously using an acoustic Doppler velocimeter (ADV) and pressure sensor mounted 1.5 m below mean water level. The layout of the tower is shown in Fig. 2. The air–sea flux data were recorded using a battery operated data acquisition system, the memory of which had to be manually downloaded to a laptop computer at regular intervals. This limited the amount of data collected during the 10-day experimental period because of practical constraints on servicing the system.

A buoy of the Chesapeake Bay Observing System (CBOS) was located about 2 km southwest of the tower at 38°28′24″N, 76°22′48″W (Fig. 1). The buoy recorded wind speed and direction, water and air temperatures, relatively humidity, barometric pressure, salinity, and current speed and direction continuously over the duration of the experiment. A wind sensor at z = 3.5 m recorded average wind speed and direction every half hour. Surface currents were measured 2.4 m below the water surface. Barometric pressure data from the buoy were utilized to correct the underwater pressure data collected at the tower. The buoy surface water temperature (Tw) was used for atmospheric stability corrections. The surface current data were used to correct surface wind speed.

b. Air–sea fluxes

The Gill sonic anemometer and IRGA package on the fixed tower recorded the turbulent Reynold stress ρauw′〉, wind speed and direction, air temperature (Ta), and heat and moisture fluxes. The package sampled at 10 Hz for 30-min intervals, collecting a total of 18 000 data points during each interval. It resolved the wind into three orthogonal components, u along the direction of the mean wind, υ horizontally transverse to it, and w in the vertical. Eddy fluxes of momentum, heat, and moisture were calculated by combining the hygrometer measurements of water vapor and Ta with the sonic anemometer measurements, using the micrometeorological methods of McMillan (1987) and Baldocchi et al. (1988). Momentum flux from air to water was calculated as
i1520-0485-32-11-3058-e12
Eddy flux of sensible heat, H, was computed as
i1520-0485-32-11-3058-e13
where ca is the specific heat of air at constant pressure. Latent heat flux, LE, was calculated as
i1520-0485-32-11-3058-e14
where LE is the evaporation rate, Lvap is the latent heat of vaporization of water, and q is the specific humidity. Here L can be calculated as
i1520-0485-32-11-3058-e15
Wind speeds measured at the tower were converted to Un10 using the calculated stability length from the flux measurements; Un10 was corrected with surface currents by
Un10un10uc
where un10 is the neutral 10-m wind velocity before correction and uc is the surface current.

We used 30 min as the averaging time both for sampling and calculating the covariances. Under stationary and horizontally homogenous conditions, the averaging time used to define the perturbations can be as small as 5 min to capture most of the stress value. But the flux average time should be much longer. Pierson (1983) suggested an averaging time of 20 min for air–sea turbulent studies. However, Donelan (1990) found that a 20-min averaging time produced rather inaccurate stress estimates, and the accuracy was particularly poor in light winds at z < 10 m. Banner et al. (1999) pointed out that data must be measured over sufficiently large time and space scales to capture all scales of variability for the eddy correction technique. The 30-min averaging time at z = 3.8 m was chosen to reduce sampling errors in the data analysis, especially since the dataset contained about 25% data points in light winds. This is also long enough to include all the significant spectral frequency contributions from surface waves (Geernaert et al. 1988).

c. Surface waves

Surface waves were measured by a 5-MHz Sontek ADV with a pressure sensor. The instrument recorded pressure and velocity components. Both the ADV and the pressure sensor were sampled at 4 Hz for 450 s, with a new burst recorded every 30 min. Water level variations and directional wave spectra were calculated from the velocity and pressure data. A correction for changes in atmospheric pressure was first applied to the subsurface pressure record using barometric pressure data from the buoy. A modified version of WavePro, a software package developed by Woods Hole Instrument Systems, Ltd., was used to carry out the directional wave spectral analysis. WavePro uses the directional spectral analysis technique of Longuet-Higgins (1963) to calculate directional wave spectra. The modification made was that the subsurface pressure and velocity record were transferred to surface elevation using a semi-empirical transfer function in the time domain (Nielsen 1989), instead of the standard linear transformation that is made after the signal is converted to the frequency domain. Nielsen's procedure was used because linear wave theory is not expected to perform well at high frequencies when calculating surface elevation from subsurface pressure, largely because of instrument noise. In this dataset, the high frequency information up to the cutoff Nyquist frequency is important and needs to be retained. Nielsen's function was designed by direct physical reasoning supplemented as appropriate with empirical quantification in the time domain. Significant wave heights, peak periods and mean wave directions were derived from the directional wave spectra.

The measured wind, surface wave, and air–sea temperature data are shown in Fig. 3. The CBOS raw wind vectors are plotted in Fig. 3. The tower wind data are not shown because they were not a continuous time series. The available air–sea flux data are shown in Fig. 3 by dots under the raw wind vectors.

d. The frontal systems

During the summer season, frontal systems frequently pass through the study area. We inspected the NOAA Daily Weather Map (weekly series) for the period of 19–28 July 1998 when the tower experiment was conducted. A day before the tower was deployed, there was a strong cold front passing through the area. On 19 July, this front became stationary and stalled far south of the area, near the bay mouth. It moved away the next day. The wind and Ta recorded from the buoy did not show another clear frontal passage until 24 July when a strong cold frontal system formed. Both wind and Ta in Fig. 3 showed the influence of this front with wind direction veering from southwest to northwest and Ta dropping significantly. On 25 July, a low pressure center became established east of Virginia and the frontal system became stationary south of CB for the next three days until 29 July.

e. Data selection

The succession of frontal systems during the field experiment made the study area less stationary and spatially homogenous and might have led to errors in the air–sea flux measurements (Ataktürk and Katsaros 1999). Therefore, a careful selection of the measured data was carried out to reduce the influence of rapid spatial and temporal variability. The potential influence of large-scale flows on air–sea momentum fluxes was recently addressed in the Southern Ocean Waves Experiment (SOWEX; Banner et al. 1999; Chen et al. 2001), but this topic is beyond the scope of the present study.

In order to focus on the role of surface waves on the air–sea momentum fluxes, we carefully selected the available air–sea flux data to meet the stationarity and horizontal homogeneity required by the Monin–Obukhov similarity theory. If the wind speed was extremely low (<1 m s−1) or there were rapid changes in wind direction and speed, that half-hour's data were not used. We also compared wind data measured by the sonic anemometer at the tower site with those simultaneously measured at the CBOS buoy. If the difference in wind speed was more than 1 m s−1 or the difference in the wind direction was greater than 45°, that half hour's data were not used either. Because similarity theory is considered valid only if stability parameters (z/L) are in the range of [−1, 1] (Valigura 1995), both extremely stable or unstable conditions with z/L > 1 or z/L < −1 were further excluded from the analysis since in these cases the turbulence is either severely damped or dominated by buoyancy production. Furthermore, data from wind, wave, and current directions that would have placed either sensor in the wake of the tower were discarded to eliminate systematic errors due to wind or current shadowing. A total of 17 data points were eliminated using these criteria, all of them with Un10 less than 4 m s−1. We had 135 data points left for analysis relative to wind speeds. Among them, 34 (25%) data points had wind speed less than 4 m s−1 and 101 (75%) of them had wind speed greater than 4 m s−1.

Wave data were selected independently from the wind data. The analyzed peak wave periods had anomalies when significant wave height was very low because the response of the pressure sensor was not very reliable for high frequency waves. Therefore, all data points with significant wave heights less than 10 cm were eliminated. A total of 119 data points with both air–sea flux data and wave data were available for further analysis.

3. Results and discussion

The 135 data points for wind speed analysis were equally grouped into nine 1 m s−1 wind speed bins, from 0.5 to 1.5 m s−1, … , 8.5 to 9.5 m s−1. The mean values of Cd and Un10 were calculated for each bin. The standard error, defined as the sample standard deviation divided by the square root of the number of samples, was also calculated. The mean Cd, standard deviation, standard error, and the sample number for each wind speed bin are listed in Table 1. Also, the mean Cd with standard error bars are plotted against Un10 in Fig. 4. Both Table 1 and Fig. 4 show that Cd increases with decreasing wind speed under light winds (Un10 < 4 m s−1) and has values much higher than for relatively higher wind speeds (4 m s−1 < Un10 < 10 m s−1); Cd increases with increasing wind speed for Un10 > 4 m s−1. The standard errors for low wind speeds are significantly larger than those for higher wind speeds.

Across the entire range of smooth, transitional, and rough turbulent flow Cd can be defined by z0, as the sum of Eq. (5) and (6), such that
i1520-0485-32-11-3058-e17
Using Eq. (17), the calculated Cd for α = 0.012 is plotted in Fig. 4 for comparison to our data. The data are approximately 90% higher than this relationship at the lowest wind speeds. Thus, an increase in apparent roughness due to the growth of the viscous sublayer does not explain the magnitude of the Cd increase that we observed at low wind speeds. On the other hand, our data are about 20% less than the Charnock relationship for Un10 > 4 m s−1. We also calculated Cd corresponding to the wave-age-dependent surface roughness in Eq. (10) proposed by Donelan (1990). We used our observed values of Hs, Tp, and Un10 for these calculations and plotted the results in Fig. 4. They show less scatter than our data, are generally larger than our data for Un10 > 4 m s−1, agree with the Charnock formulation for 4 m s−1 < Un10 < 7 m s−1, and are larger than the Charnock relationship for Un10 > 7 m s−1.
In order to compare our observed dependence of Cd on Un10 with other studies, data points with Un10 > 4 m s−1 were linearly regressed as in Eq. (7) using N = 101 samples. A best fit is found to be
CdUn10−3
The correlation coefficient was quite low (r = 0.273). The linear regression in Eq. (18) is plotted in Fig. 5 for comparison with results from MARSEN, HEXMAX, RASEX, and SWADE. The Charnock relationship is also shown for comparison. Our data show a lower Cd than all of the other observations when Cd is considered as a function of Un10 alone. This discrepancy in comparison with other results and the large scatter imply that another source of variability, very possibly the wave age, may play an important role.

To test the dependence of the drag coefficient on wave age, the 119 pairs of Cd and Cp/u∗ were fit to an equation of the form of Eq. (9). Our data gave a best fit of A = 6.28 × 10−3 and B = −0.578, with a correlation coefficient of r = 0.64, significantly higher than that for the Un10 fit. The standard error for A was 1.105 × 10−3 and for B was 0.0589. The regression is plotted in Fig. 6 for comparison with MARSEN and RESEX fits to the same form. The three regression curves are quite similar to each other, with our data regression line falling between the curves from MARSEN and RESEX. In other words, our data show good agreement with the results of other studies when considered as a function of wave age, even when comparisons as a function of wind speed to those same studies showed poor agreement.

It is possible that the empirical fit between Cd and Cp/u∗ suffers from self-correlation problems because u∗ appears on both sides. Following Perrie and Toulany (1990) and Smith et al. (1992), we estimated the spuriousness introduced by the variability of u∗. Spurious correlation would be ignored if the following conditions are met:
i1520-0485-32-11-3058-e19
For our 119 data points, the estimated variances are 0.0527 < 0.126 and 0.158 < 0.333. Both conditions are not strictly met, but are approximately met. This means Cp and u∗ have a good range to avoid self-correlation problems in our dataset.

The wave height and steepness dependence of surface roughness proposed by Taylor and Yelland (2001) does not explain our data very well. We calculated the drag coefficient corresponding to Eq. (11) based on measured Hs and wave steepness for comparison to Cd from direct flux measurements. The results are presented in Fig. 7, where it is apparent that Eq. (11) underpredicted Cd severely for young waves (Cp/u∗ < 12), just as Taylor and Yelland pointed out in their study.

The temporal variability of the drag coefficient in our dataset is illustrated in Fig. 8, where an 11-h time series data segment of nearly continuous estimates of wind and drag coefficient are presented beginning at 0000 EST 23 July 1998. Here Un10 increased from less than 2 m s−1 to near 8 m s−1 in 8 hours with almost the same wind direction from southwest toward the northeast (Fig. 8b). The correlation coefficient between measured Hs and Un10 for the whole experiment was 0.78 and the waves followed the wind direction very closely (Lin et al. 2002). The calculated Cd from air–sea flux measurements shows a significantly higher value for low winds than that for higher winds (Fig. 8c). Equation (17) predicted a nearly constant Cd and Eq. (18) underpredicted the Cd over the entire period. Equation (9) predicted a Cd in a better agreement with the data. The measured and calculated Cd are further shown as a function of Un10 in Fig. 8d. Once again, Cd increases with decreasing wind speed under light winds (Un10 < 4 m s−1) and has values much higher than for relatively higher wind speeds (4 m s−1 < Un10 < 10 m s−1). The scatter for low winds is significantly larger than for higher winds.

4. Modeling air–sea interaction processes

a. Model description

Numerical models have been used previously to study wave effects on drag. Janssen (1989) used a numerical model based on resonant wave–mean flow interaction and the quasilinear theory of wind wave generation to theoretically study the effects of waves and air turbulence on the wind profile. He found that there is a strong coupling between wind and waves for young windseas and that a Cp/u∗ scaling of Cd is better than Un10. However, the iteration procedure to solve the nonlinear set of equations in Janssen's model was too time consuming to apply operationally (Janssen 1991). We used a relatively simple numerical wave model to study the same problem. The model is based on a parametric, deep water numerical wave model first developed by Donelan (1977) and revised by Schwab et al. (1984). The control equation is based on a local momentum balance. The model is time dependent and can accommodate arbitrary wind and geography. Shallow water wave effects are not included, however. The wave energy spectrum E(f) is assumed to follow the JONSWAP spectral shape (Hasselmann et al. 1973). Estimated Cd is output based on the forcing wind speeds and the predicted wave conditions at each hour.

In the wave model, the total drag τ is treated as the sum of a skin friction (or tangential stress) τs, and a wave form drag τf:τs is proportional to U2n10 as
τsρaCdsU2n10
where Cds = 0.7 × 10−3 is the skin friction coefficient, and τf is assumed to be proportional to the square of the relative velocity between the wind velocity and the wave phase velocity,
τfρaDfUn10CUn10C
This is similar to the treatment of a mobile sea surface proposed by Kitaigorodskii and Volkov (1965). The wave form drag changes with the development of waves, such that in a fully developed sea the dominant wave travels at a phase speed close to the wind velocity and τf becomes smaller and smaller. Full development corresponds to wave age, Cp/Un10 = 1.2 (Donelan 1990; Pierson and Moskowitz 1964; Bretschneider 1973). A relationship between the mean wave phase velocity C and peak wave phase velocity Cp was proposed by Bretschneider (1973) as Cp/C = 1.2. Using this relationship
τfρaDfUn10CpUn10Cp
where Df is the form drag coefficient given by
i1520-0485-32-11-3058-e23
In the original model, z0 = σ/5 was used where σ2 = 0 E(f) df is the variance of surface elevation. We tested this expression against the data and found that it predicted values of z0 one to two orders of magnitude higher than the observed surface roughness. We modified the original model to calculate z0 using Eq. (10), which explicitly accounts for the effect of wave age on surface roughness and agrees better with our data. Finally Cd is calculated as
i1520-0485-32-11-3058-e24

b. Application

The modified wave model was run for the experimental period. The geographic region of interest was represented as a horizontal grid covering the area north of 36°39′N, south of 39°36′N, east of 77°23′W and west of 75°40′30″W. The west–east direction was divided into 410 grid points with a spatial resolution of 365.3 m and a total length of 149.8 km. The south–north direction was divided into 708 grid points with a spatial resolution of 462.5 m and a total length of 327.5 km.

Hourly wind speeds from the CBOS buoy were used along with wind data from the Thomas Point Light (TPL) tower maintained by NOAA. TPL is located at 38°59′18″N, 76°26′12″W, approximately 55 km north of the tower (Fig. 1). Wind speed and direction were measured hourly at z = 18 m. Wind speeds were converted to Un10 according to the following procedure. An initial guess of u∗ is calculated, based on Cd = 1.2875 × 10−3 (Wu 1982), when Un10 < 7.5 m s−1, and Cd = (0.8 + 0.065 × Un10) × 10−3 (WAMDI Group 1988), when Un10 ≥ 7.5 m s−1. For this initial guess, Uz is used instead of Un10. The friction velocity u∗ is then estimated. If the boundary layer is neutral, the measured wind Uz only needs to be adjusted to 10-m height using similarity theory. If the boundary layer is stable or unstable, then L in Eq. (2) is first calculated without stability correction [assuming ψm(z/L) = 0]. An iterative process is then started to calculate the stability parameter in Eq. (3) or (4) with the initially estimated L. After L was recalculated in Eq. (2) with the stability correction, the stability parameter in Eqs. (3) or (4) could be recalculated as well. A new u∗ was then estimated. This process was repeated until the u∗ estimates converged (Δu∗ ≤ 10−6). Then Un10 was interpolated linearly over latitude for model input. The model determined the time step dynamically based on maximum winds for each hour. The time step ranged from 0.1 to 3 min; Hs, Tp, and Cd were output every hour for further analysis.

c. Results

The model-predicted Hs and Tp are plotted against the measured data in Fig. 9 for measured Hs ≥ 10 cm (total 129 data points). Figure 9 shows that the wave model predicted Hs and Tp reasonably, though with a fair amount of scatter. The scatter index (SI) is defined as the root-mean-square error normalized by the mean observed value of the reference quantity. It is found that SI = 0.58 for Hs and SI = 0.36 for Tp.

The 220 modeled Cd are grouped into nine 1 m s−1 interval wind speed bins. The mean values of Cd, standard deviation, standard error, and the sample number for each wind speed bin are listed in Table 1. Bin-averaged predicted Cd with standard error bars is also plotted against mean wind speed in Fig. 10 for comparison to the bin-averaged data. In comparison with the data, the model standard errors are significantly smaller, as expected. Similar to the data, however, the standard errors in light winds are much larger than those for higher winds. Modeled Cd increased with decreasing wind speed under light winds (Un10 < 3 m s−1), with higher values than those at higher winds (Un10 ≥ 3 m s−1). This was the same behavior as the data except that the transitional wind speed from the modeled Cd occurred at Un10 = 3 m s−1 rather than 4 m s−1. Modeled Cd was higher than the data for Un10 ≥ 4 m s−1, but lower for Un10 < 3 m s−1. Model-predicted Cd was a much better approximation to the data at low wind speeds than any of the other formulations tested, even though the model did not include the effects of increasing viscous sublayer thickness on τs.

The 127 modeled data points with Un10 ≥ 4 m s−1 are also linearly fit to Un10. The best fit was found to be
CdUn10−3
with a correlation coefficient of r = 0.73, which is significantly larger than that calculated from data. The scatter of the modeled Cd is reduced significantly in comparison to the data, but it is not eliminated. The scatter in Hs and Tp predictions contributes to the scatter of Cd through Df. The model also predicts 20% higher Cd than that derived directly from the data, and it matches the MARSEN, HEXMAX, and SWADE results.

Model results were also analyzed to fit estimated Cd to estimated wave age using Eq. (9). The best fit gave A = 6.79 × 10−3 and B = −0.592, with r = 0.82, a standard error for A of 5.606 × 10−4 and for B of 0.027. All of these error indications are significantly smaller than those from the data. The two regression lines from model and data are plotted in Fig. 11 for comparison to the individual model predictions. There is a close agreement between these two lines. The statistical behaviors of the model predictions and the data are very similar when considered as a function of wave age.

d. Discussion

The fact that a relatively simple, momentum-driven deep water wave model did so well at predicting the behavior of the air–sea drag coefficient in Chesapeake Bay is rather remarkable. These predictions were particularly good when considered as a function of wave age, which adds support to the argument that wave age is the dominant control on the air–sea drag coefficient. In a recent laboratory study of the wind stress partition between wave form drag and viscous tangential stress, Banner and Peirson (1998) reported that for wind speed up to 14 m s−1 the tangential stress coefficient decreases with wind speed and is relatively insensitive to the sea state. As the overall drag coefficient increases with wind speed, the relative importance of the form drag also increases. According to Fig. 2 in Banner et al. (1999), Cds is in the range of 0.7–0.9 (×10−3) for Un10 = 6 m s−1 and 0.6–0.8 (×10−3) for Un10 = 10 m s−1. In our datasets, the maximum Un10 = 9.34 m s−1, the constant value of Cds = 0.7 × 10−3 adopted for simplicity in our model (Donelan 1977) is a good approximation in this range.

As observed by Banner and Peirson (1998), the model also predicted an increasing Cd with Un10 (see Fig. 10) through the increasing form drag coefficient, Df with Un10. When waves become fully developed, they no longer contribute to the surface roughness length both because they are not steep and because they travel at speeds equal or higher than the wind speeds (Donelan 1990). When the peak phase speed reaches 1.2Un10, the wave form drag in Eq. (22) diminishes with Df being a constant. This means that in a fully developed wave field, the dominant waves traveling at phase speeds close to the wind velocity receive little momentum from air. Since fully developed waves occur in high wind speeds, the stratification effects are not important because Lu3 becomes large and since L → ∞, z/L → 0, and ψm(0) = 0. The similarity theory predicts that the atmospheric boundary layer can be assumed neutrally stratified.

5. Summary and conclusions

In this study, we present data from an atmospheric boundary layer and surface wave study in CB under almost purely fetch-limited conditions, with fetches ranging from a few to tens of kilometers. The analyzed tower data provided us an opportunity to examine the properties of the air–sea drag coefficient for fetch-limited wind waves in low wind conditions. The dataset was unique in that it contained simultaneously measured air–sea fluxes, surface wind waves, and surface current data. The data were obtained in a fetch-limited, semienclosed basin dominated by windseas with very little swell. The analyzed results lead to the following conclusions:

The neutral drag coefficients depend upon both wind speed and wave age, but they are better correlated to wave age than wind speed. When Un10 < 4 m s−1, the neutral drag coefficients are found to increase with decreasing wind speed and have values much higher than for relatively higher wind speeds. When 4 m s−1Un10 < 10 m s−1, the neutral drag coefficients increase with increasing wind speed. The data show significantly lower drag coefficients in comparison with other datasets. The data scatter is significantly larger in low winds than in high winds. Regardless of the wind speed, the neutral drag coefficients always decrease with increasing wave age. Regression of the neutral drag coefficients with wave age results in a much higher correlation coefficient than regression with wind speed, and the results agree with those of other studies.

We used a sea-state-dependent surface roughness and a reference system moving with the waves in a parametric numerical wave model to predict wave-age-dependent drag coefficients. The model-predicted drag coefficients reproduce the wind speed dependence of the data qualitatively, though predicted drag coefficients at low wind speeds are slightly lower than the data and predicted drag coefficients at high wind speeds are slightly higher than the data. The modeled drag coefficients reproduce the wave age dependence of the data very well. This suggests that the momentum transfer mechanism implied by wave-age-dependent drag can be physically explained by a wave-age-dependent form drag coefficient with a dependence on the relative velocity between wind and waves, rather than the absolute wind velocity. The wave model used in this study, which predicts total drag as a byproduct of the wave calculation and is computationally fast, is a good tool for coupling with mesoscale atmospheric models to effect improved, fully coupled estimates of winds, waves, and air–sea fluxes of heat and momentum.

Acknowledgments

We are grateful to Dr. William Boicourt for making space available to us on the tower and for giving us access to the CBOS wind data. We thank Drs. Mark Donelan and David Schwab for sharing their source code and providing guidance on its use. We also thank two anonymous reviewers for their helpful comments. This study was supported by the National Sea Grant Office (Grant NA86RG0037) and by the National Ocean Partnership Program (Grant N00014-98-1-0837).

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Fig. 1.
Fig. 1.

The Chesapeake Bay bathymetry and the tower and buoy locations for the field experiment: ★ tower location, ▴ CBOS buoy, and + Thomas Point Light tower

Citation: Journal of Physical Oceanography 32, 11; 10.1175/1520-0485(2002)032<3058:DCWFLW>2.0.CO;2

Fig. 2.
Fig. 2.

Layout of the temporary fixed tower for the experiment: (a) cross section and (b) plan view

Citation: Journal of Physical Oceanography 32, 11; 10.1175/1520-0485(2002)032<3058:DCWFLW>2.0.CO;2

Fig. 3.
Fig. 3.

The time series of (a) measured wind speeds and directions, (b) significant wave heights and mean wave directions, (c) peak wave periods, and (d) air and sea temperatures for the tower experiment from 19 to 28 Jul 1998. The lengths of the vectors are used to show the magnitude of the wind speed or significant wave height. The pointing directions of the vectors show wind directions or mean wave directions. The wind data in (a) are from CBOS raw wind data with the available air–sea flux data shown by dots under the raw wind vectors

Citation: Journal of Physical Oceanography 32, 11; 10.1175/1520-0485(2002)032<3058:DCWFLW>2.0.CO;2

Fig. 4.
Fig. 4.

Comparison of observed mean neutral drag coefficients for nine 1 m s−1 different wind speed bins with those calculated from Eq. (17) (dotted line) and Eq. (10) (solid squares). The data points are shown by open circles. Error bars represent the standard error for each wind speed bin

Citation: Journal of Physical Oceanography 32, 11; 10.1175/1520-0485(2002)032<3058:DCWFLW>2.0.CO;2

Fig. 5.
Fig. 5.

Linear regression between the measured neutral drag coefficients and the mean wind speeds, Un10, as compared with results from other field studies. Data points are shown by open circles. Lines from the top to the bottom are: Charnock (short dash), RASEX (dash-dotted), MARSEN (dotted), HEXMAX (long-dash), SWADE (dash-dotted-dotted), and regression from data by Eq. (18) (solid)

Citation: Journal of Physical Oceanography 32, 11; 10.1175/1520-0485(2002)032<3058:DCWFLW>2.0.CO;2

Fig. 6.
Fig. 6.

Nonlinear regression between measured neutral drag coefficient and wave age in Eq. (9). Data points are shown by open circles. Results from the present study (solid line) is in between those from MARSEN (dotted line) and RASEX (dash-dotted line)

Citation: Journal of Physical Oceanography 32, 11; 10.1175/1520-0485(2002)032<3058:DCWFLW>2.0.CO;2

Fig. 7.
Fig. 7.

Comparison of the measured neutral drag coefficients from the flux measurement (y axis) with those calculated by Eq. (11) (x axis) (Taylor and Yelland 2001). Solid circles: Cp/u∗ < 12, solid triangles: 12 < Cp/u∗ < 20, and open circles: Cp/u∗ > 20

Citation: Journal of Physical Oceanography 32, 11; 10.1175/1520-0485(2002)032<3058:DCWFLW>2.0.CO;2

Fig. 8.
Fig. 8.

An 11-h time series of (a) neutral wind speed, (b) wind direction, and (c) the drag coefficients. The Cd in (c) was calculated from flux measurements (open circles), from linear regression in Eq. (18) (solid lines), from the regression with wave age in Eq. (9) (solid triangles), and from Eq. (17) (dotted line). These Cd are plotted against the mean wind speeds Un10 in (d)

Citation: Journal of Physical Oceanography 32, 11; 10.1175/1520-0485(2002)032<3058:DCWFLW>2.0.CO;2

Fig. 9.
Fig. 9.

(a) The wave model–predicted significant wave height Hs (y axis) vs the measured Hs > 10 cm (x axis) and (b) the model-predicted peak period Tp (y axis) vs measured Tp (x axis)

Citation: Journal of Physical Oceanography 32, 11; 10.1175/1520-0485(2002)032<3058:DCWFLW>2.0.CO;2

Fig. 10.
Fig. 10.

Comparison of the model-predicted drag coefficients for nine 1 m s−1 wind speed bins with the observed mean drag coefficients. Model-predicted Cd are shown by open triangles. The dash-dotted line with solid triangles shows the model-predicted bin-averaged Cd. The error bars represent standard errors for each bin. The bin-averaged data are shown by solid line with solid circles

Citation: Journal of Physical Oceanography 32, 11; 10.1175/1520-0485(2002)032<3058:DCWFLW>2.0.CO;2

Fig. 11.
Fig. 11.

The regression line calculated from the model-predicted drag coefficients (dash-dotted line) with the model-predicted wave age is compared with that from the measured data (solid line). Model-predicted drag coefficients are shown by open triangles

Citation: Journal of Physical Oceanography 32, 11; 10.1175/1520-0485(2002)032<3058:DCWFLW>2.0.CO;2

Table 1. 

The neutral drag coefficients for different wind speed bins from data and model

Table 1. 

* University of Maryland Center for Environmental Science Publication Number 3577.

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